# 4 Integral calculus

We now turn from differentiation to the other crucial ingredient of the infinitesimal calculus, namely *integration*. This may be approached in two ways: either as the inverse process of differentiation; or as the means of calculating the area under a given curve, using an argument that serves as a template for many other important applications. However, before discussing these, we must develop the above two approaches, and the relations between them.

## 4.1 Indefinite integrals

Given a function *f*(*x*), the *indefinite integral* *F*(*x*) is defined as the most general solution of the equation

It is not unique. Suppose we have a particular solution *F*_{0}(*x*), with

Then the most general solution can be written

where *G*(*x*) is any function such that (4.1) is satisfied. On substituting (4.3) into (4.1) and using (4.2), one obtains , and so *G*(*x*) = *c*, where *c* is a constant. Hence the indefinite integral is given by

where *F*_{0}(*x*) is any particular solution ...

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